Geodesic
A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics
The electronic journal of combinatorics, Tome 13 (2006)
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A "hybrid method", dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.
DOI : 10.37236/1129
Classification : 05A16, 05A15, 30B10, 33B30, 40E10
Mots-clés : generating functions, asymptotic expansions 
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     author = {Philippe Flajolet and Eric Fusy and Xavier Gourdon and Daniel Panario and Nicolas Pouyanne},
     title = {A hybrid of {Darboux's} method and singularity analysis in combinatorial asymptotics},
     journal = {The electronic journal of combinatorics},
     year = {2006},
     volume = {13},
     doi = {10.37236/1129},
     zbl = {1111.05006},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1129/}
}
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Philippe Flajolet; Eric Fusy; Xavier Gourdon; Daniel Panario; Nicolas Pouyanne. A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics. The electronic journal of combinatorics, Tome 13 (2006). doi: 10.37236/1129

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